Description
I will present homotopy continuation methods for solving 0-dimensional polynomial systems, where each polynomial is expressed as a general linear combination of prescribed, fixed polynomials. This approach involves selecting a specific starting system for homotopy continuation, leveraging the theory of SAGBI (Khovanskii bases) and toric geometry. For square systems, SAGBI homotopies can significantly reduce the number of solution paths to track compared to polyhedral homotopies, which are currently the default in HomotopyContinuation.jl.
As a direct application of this theory, I will demonstrate the SAGBI homotopy with two examples: (1) finding approximate stationary states for coupled driven nonlinear resonators — a problem in nonlinear dynamics, and (2) computing approximate solutions to the electronic Schrödinger equation in coupled cluster theory, which arises in quantum chemistry.
